How come this natural logarithm approximation works?

[Andreas C]

I came across a guy claiming that the "best approximation" for the natural logarithm of a number is this:
ln x=2^n*(x^(2^-n)-1)

Oddly enough, it seems to work rather well! I don't really get why it does... I also don't know if it has a limit, I couldn't test it as I don't have access to my laptop right now, but I can't understand WHY it seems to work, at least for n as high as 25.

 

[mfb]

What is n?

With n=0, we get $\ln(x) \approx x-1$ which is the start of the taylor series around x=1.

In the limit n->infinity, the formula is exact. Introduce u=2^n to shorten formulas:

$$\lim_{u \to \infty} u (x^{1/u}-1) = \lim_{u \to \infty} u (e^{ln(x)/u}-1) = \lim_{u \to \infty} u (1+\frac{\ln(x)}{u} + O(\frac{1}{u^2})-1) = \ln(x)$$

 

[Andreas C]

N is a very large natural number.

 

[Andreas C]

I tested it with n=25 for 2, 5, 10 and 15 and for some weird reason it worked.

 

[RUber]

Is this a series? Where $\ln x = \sum_{n = 0}^\infty 2^n \left( x^{2^{-n}} - 1 \right) ?$
If so, compare that to the power series of the log centered at 2.
$\ln x = \ln 2 + \sum_{n = 1}^\infty \frac{(-1)^{n-1} (x - 2)^n}{2^n n}$
It looks like there might be some similar terms...

 

[Andreas C]

First, I think you forgot something, the first term in the parenthesis is x^(2^-n), not x^(2^n).

No, it's not a series.

 

[mfb]

See my edit in my previous post. For the limit of n to infinity, the formula is exact (proof in the post). For large n, it can give a reasonable approximation, the range where the approximation works grows with n.

 

[Andreas C]

See my edit in my previous post. For the limit of n to infinity, the formula is exact (proof in the post). For large n, it can give a reasonable approximation, the range where the approximation works grows with n.

Oh, so it is actually a good approximation. Weird, I had never heard of it again. Is anyone aware of where it first appeared?

 

[mfb]

It is prone to numerical errors, for large n you subtract 1 from a number close to 1. I don't see practical applications of this formula in this direction. Maybe in the opposite direction (using log(x) to approximate the exponential), but that is equivalent to a taylor approximation.

 

[Andreas C]

It is prone to numerical errors, for large n you subtract 1 from a number close to 1. I don't see practical applications of this formula in this direction. Maybe in the opposite direction (using log(x) to approximate the exponential), but that is equivalent to a taylor approximation.

Yes, it's not practical, but it's accurate if you're accurate. At first I thought it was something like a mathematical coincidence that only sometimes works, but it turns out that it actually works.

On a slightly unrelated note, do you know of any such "coincidences" or patterns or rules that only start to break after many, many numbers? I've always thought they were interesting mathematical objects, functions or series that "cheat" you into thinking they are going to be a certain way and then change unpredictably.

 

[mfb]

The Borwein integral
The Hirsch conjecture fails first in 43 dimensions.

Pólya conjecture: Is there a number N such that less than half of all numbers smaller than N have an odd number of prime factors?
Yes, but the smallest example is 900 millions.

a=n²⁹ + 14 and b=(n+1)²⁹ + 14 are relative prime for all numbers smaller than n=345253422116355058862366766874868910441560096980654656110408105446268691941239624255384457677726969174087561682040026593303628834116200365400, a number with 141 digits. Taken from here.

Does the number of primes below N exceed $\int_0^N \frac {dt}{\ln(t)}$ for some N? Yes, but the smallest example must be larger than 10¹⁹, and it is expected to be around 10³¹⁶. Related to Skewes' number.

Mertens conjecture, yet another one about prime factors. The smallest counterexample has to be above 10¹⁶, but it could be as large as ~10¹⁰⁴⁰.

 

[fresh_42]

So there is still hope for my vicious fantasy that ERH will fail somewhere.
Shouldn't your and my response be better placed in the Collatz conjecture thread?

 

[Andreas C]

So there is still hope for my vicious fantasy that ERH will fail somewhere.
Shouldn't your and my response be better placed in the Collatz conjecture thread?

What is "ERH"? Anyways, I think that means there is still hope Collatz conjecture may be proven wrong!

 

[fresh_42]

Yes, and ERH is the extended Riemann hypothesis, which would be a sensation if proven wrong. I found @mfb's examples very interesting. I've read of an example about primes before, which first fails at $41$ and thought this would be a large number. I couldn't be more wrong. So thanks for this thread, which brought it up.

 

[MAGNIBORO]

Yes, it's not practical, but it's accurate if you're accurate. At first I thought it was something like a mathematical coincidence that only sometimes works, but it turns out that it actually works.

On a slightly unrelated note, do you know of any such "coincidences" or patterns or rules that only start to break after many, many numbers? I've always thought they were interesting mathematical objects, functions or series that "cheat" you into thinking they are going to be a certain way and then change unpredictably.

Yes, it's not practical, but it's accurate if you're accurate. At first I thought it was something like a mathematical coincidence that only sometimes works, but it turns out that it actually works.

On a slightly unrelated note, do you know of any such "coincidences" or patterns or rules that only start to break after many, many numbers? I've always thought they were interesting mathematical objects, functions or series that "cheat" you into thinking they are going to be a certain way and then change unpredictably.

trigonometrics integrals like The Borwein integral
http://www.jstor.org/stable/40378359?seq=2#page_scan_tab_contents
you must be log in , page 282 And onwards